Here's my take: by the time I finished writing in the candidates, 24 cells had been solved. A scan of rows, columns and boxes solved five more. There was a Type 4 rectangle. However, I spotted no more rectangles or basic stuff and no XY-Wings were apparent, so I turned to looking for strong links and noting them outside the grid.

There were no X-Wings to come out of this, but there was a strong link (fork? skyscraper?). This strong link solved a cell, which in turn solved the puzzle. It's rather unusual for me to see a strong link make such a powerful impact.

I like that you post some of these puzzles. They're a welcome change of pace from the same old puzzles that I do on a day-in, day-out basis.

The UR on <69> is there in R35C78. It eliminates <9> from R3C78, which is not much help.

Like you (I think) I found the strong links on <5> to be interesting. In particular, the strong links in R37 line up in C7. So the "fork" takes out the possibility <5> in R8C2, and the puzzle is solved.

There is another neat thing. Look at C19. Almost an X-wing on <5>. Except for the candidate <5> in R7C1. I believe this is a "finned" X-wing. Anyway, here is my reasoning:

Either the X-wing is true, or R7C1 = <5>.

If the X-wing is true, R9C5 is not <5>.
If R7C1 = <5>, R9C9 = <5>, and R9C5 is not <5>.

Then, you have an X-wing in R19, and R7C1 is not <5>.

Good stuff!

(If you do not exploit this coloring on <5>, the puzzle seems to be quite difficult to solve.)

Like you (I think) I found the strong links on <5> to be interesting. In particular, the strong links in R37 line up in C7. So the "fork" takes out the possibility <5> in R8C2, and the puzzle is solved.

When I saw the fork, I immediately took out the "5" from r1c1, having not noticed that it could also solve r8c2. But, of course, the effect is the same.

Quote:

There is another neat thing. Look at C19. Almost an X-wing on <5>. Except for the candidate <5> in R7C1. I believe this is a "finned" X-wing. Anyway, here is my reasoning:

Either the X-wing is true, or R7C1 = <5>.

If the X-wing is true, R9C5 is not <5>.
If R7C1 = <5>, R9C9 = <5>, and R9C5 is not <5>.

Then, you have an X-wing in R19, and R7C1 is not <5>.

Naturally, I didn't get that far, since the fork precluded the need to look for strong links in columns. I just learned the finned X-Wing a few months ago from the guys on this forum.

However, my understanding of that technique is that it removes one or more candidates from the box in which the fin occurs, because those candidates would be gone regardless of whether the X-Wing existed or the fin prevented it. In this case, the finned X-Wing couldn't be used because there was no candidate in the box that would have been eliminated had the X-Wing actually existed.

What you did, as I saw it, was to solve r9c5 with a DIC. The finned X-Wing technique was unable to eliminate anything, but the fin planted the seed in your brain that said, "this might be a good place to start a DIC."

I've actually called this a colored x-wing, which is very closely related to a finned x-wing. It uses grouped coloring or multicoloring as in your case.

X and x are conjugate colors, with X combining with the starred cells in c9 to force an x-wing. A and a are also conjugate colors, and since x and a exclude each other, we know that either X, A, or both must be true. In any case, r9c5 cannot be a five.

This was a case where regular multicoloring (or strong links) also works, but there are some other examples where colored x-wing interactions and box-box interactions work when simpler coloring methods don't.

I never understood David's DIC's, because I could not see any rationale for how to look for them. If you now tell me they are based on strong links, I will have to go back for remediation! Maybe dcb can weigh in here.

What I do think is that these strong links are fundamental, especially for paper & pencil solvers like you and me. I don't think we need more names for techniques, but the idea that we can overlay strong links on other ideas (here, an X-wing) can be very useful.

As we have previously seen, with the overlay of strong links on Unique Rectangles.

I never understood David's DIC's, because I could not see any rationale for how to look for them. If you now tell me they are based on strong links, I will have to go back for remediation!

I'm certainly not telling you that. Since I registered here in February, you have been one of my valued mentors and I don't think I'm in a position to try and tell you anything, other than to relate my thoughts based on what I see and to ask questions hoping to learn something.

The only rationale I use for starting chains, if indeed it's even considered a rationale, is to eyeball the potential starting point to see if both possibilities can eliminate something, in which case I pursue it. I'd also be interested in hearing how David finds his DICs.

I will let David speak for himself on whether DICs ALWAYS entail strong links, but I will tell you that they (at least) very often do, and this is how I would have "discovered" this solution.

In looking at the set of <5> , I would note that it is a great candidate for holding a solution. The <5> is only a candidate twice in rows 1, 3, 7, and 8; columns 2, 8, and 9; boxes 1, 3, 8, and 9.

Conversely, <5> is a candidate three times ONLY in row 1, column 9, and box 7. Quite literally, then, all roads point to box 7.

In my admittedly limited experience, I all but know that not all the fives are going to fit, and box 7 holds the key, and in short order I see that if r7c1 is a <5> then r1c1 is a <5> and, consequently and in any event, r5c9 and r2c8 can't be <5>.

This is where David comes in and points out that it's just two connected DICs and then set out the logical equation...

At least I think that's how it works.

Last edited by AZ Matt on Mon Aug 28, 2006 10:42 pm; edited 1 time in total